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Unformatted text preview: The General Solution to d~x dt = A~x ( t ) is ~x ( t ) = c 1 e 1 t ~v 1 + c 2 e 2 t ~v 2 where 1 and 2 are the eigenvalues of A ~v 1 and ~v 2 are the associated eigenvectors If an initial condition is specified, ~x (0) = c 1 ~v 1 + c 2 ~v 2 . So c 1 and c 2 are just the coordinates of ~x (0) with respect to the eigenbasis ~v 1 and ~v 2 . Note : We do not consider the case when 1 = 2 (even if A is still diagonalizable) If the eigenvalues are complex: ( = p qi with eigenvectors ~v i~w ) The above equation gives all complex solutions to the system. Say p + iq is an eigenvalue with associated eigenvector ~v + i~w . All real solutions of the system are given by ~x ( t ) = e pt ~w ~v cos( qt )- sin( qt ) sin( qt ) cos( qt ) ~w ~v- 1 ~x (0) Sketching the Phase Portrait Real Eigenvalues: (see diagram on page 407 in the textbook) 1. Draw the x 1- x 2 plane. 2. Draw the straight line which is the eigenspace for each eigenvalue. If the eigenvalue is negative, trajectories tend toward the origin and if the eigenvalue is positive, they...
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